THE THREE SHEEP.
(
Chessboard Problems)
A farmer had three sheep and an arrangement of sixteen pens, divided off
by hurdles in the manner indicated in the illustration. In how many
different ways could he place those sheep, each in a separate pen, so
that every pen should be either occupied or in line (horizontally,
vertically, or diagonally) with at least one sheep? I have given one
arrangement that fulfils the conditions. How many others can you find?
Mere reversals and reflections must not be counted as different. The
reader may regard the sheep as queens. The problem is then to place the
three queens so that every square shall be either occupied or attacked
by at least one queen--in the maximum number of different ways.
Answer:
The number of different ways in which the three sheep may be placed so
that every pen shall always be either occupied or in line with at least
one sheep is forty-seven.
The following table, if used with the key in Diagram 1, will enable the
reader to place them in all these ways:--
+------------+---------------------------+----------+
| | | No. of |
| Two Sheep. | Third Sheep. | Ways. |
+------------+---------------------------+----------+
| A and B | C, E, G, K, L, N, or P | 7 |
| A and C | I, J, K, or O | 4 |
| A and D | M, N, or J | 3 |
| A and F | J, K, L, or P | 4 |
| A and G | H, J, K, N, O, or P | 6 |
| A and H | K, L, N, or O | 4 |
| A and O | K or L | 2 |
| B and C | N | 1 |
| B and E | F, H, K, or L | 4 |
| B and F | G, J, N, or O | 4 |
| B and G | K, L, or N | 3 |
| B and H | J or N | 2 |
| B and J | K or L | 2 |
| F and G | J | 1 |
| | | ---- |
| | | 47 |
+------------+---------------------------+----------+
This, of course, means that if you place sheep in the pens marked A and
B, then there are seven different pens in which you may place the third
sheep, giving seven different solutions. It was understood that
reversals and reflections do not count as different.
If one pen at least is to be _not_ in line with a sheep, there would be
thirty solutions to that problem. If we counted all the reversals and
reflections of these 47 and 30 cases respectively as different, their
total would be 560, which is the number of different ways in which the
sheep may be placed in three pens without any conditions. I will remark
that there are three ways in which two sheep may be placed so that every
pen is occupied or in line, as in Diagrams 2, 3, and 4, but in every
case each sheep is in line with its companion. There are only two ways
in which three sheep may be so placed that every pen shall be occupied
or in line, but no sheep in line with another. These I show in Diagrams
5 and 6. Finally, there is only one way in which three sheep may be
placed so that at least one pen shall not be in line with a sheep and
yet no sheep in line with another. Place the sheep in C, E, L. This is
practically all there is to be said on this pleasant pastoral subject.
